Answer :
Let's solve this step-by-step.
We need to determine which of the options is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
Option A: [tex]\(\frac{24}{30}\)[/tex]
First, simplify [tex]\(\frac{24}{30}\)[/tex]. This fraction simplifies to [tex]\(\frac{4}{5}\)[/tex] by dividing both the numerator and the denominator by their greatest common divisor, which is 6. However, [tex]\(\frac{4}{5}\)[/tex] is not raised to any power here, so it is incorrect.
Option B: [tex]\(\frac{4^6}{5}\)[/tex]
This option simplifies to a fraction where the numerator is [tex]\(4\)[/tex] raised to the power of 6 and the denominator is 5, which is not the same as [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], because we need both the numerator and the denominator raised to the power of 6 for them to match.
Option C: [tex]\(\frac{4^6}{5^6}\)[/tex]
This option matches the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], because when you raise a fraction to a power, both the numerator and the denominator are raised to that power. Therefore, [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is indeed equal to [tex]\(\frac{4^6}{5^6}\)[/tex].
Option D: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
This option simply multiplies [tex]\(\frac{4}{5}\)[/tex] by 6, which is not related to raising [tex]\(\frac{4}{5}\)[/tex] to the power of 6.
After reviewing all options, the correct answer is:
C. [tex]\(\frac{4^6}{5^6}\)[/tex].
We need to determine which of the options is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
Option A: [tex]\(\frac{24}{30}\)[/tex]
First, simplify [tex]\(\frac{24}{30}\)[/tex]. This fraction simplifies to [tex]\(\frac{4}{5}\)[/tex] by dividing both the numerator and the denominator by their greatest common divisor, which is 6. However, [tex]\(\frac{4}{5}\)[/tex] is not raised to any power here, so it is incorrect.
Option B: [tex]\(\frac{4^6}{5}\)[/tex]
This option simplifies to a fraction where the numerator is [tex]\(4\)[/tex] raised to the power of 6 and the denominator is 5, which is not the same as [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], because we need both the numerator and the denominator raised to the power of 6 for them to match.
Option C: [tex]\(\frac{4^6}{5^6}\)[/tex]
This option matches the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], because when you raise a fraction to a power, both the numerator and the denominator are raised to that power. Therefore, [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is indeed equal to [tex]\(\frac{4^6}{5^6}\)[/tex].
Option D: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
This option simply multiplies [tex]\(\frac{4}{5}\)[/tex] by 6, which is not related to raising [tex]\(\frac{4}{5}\)[/tex] to the power of 6.
After reviewing all options, the correct answer is:
C. [tex]\(\frac{4^6}{5^6}\)[/tex].
